Simplified Design of Controllers to Solve the Robust Decentralized Servomechanism Problem for Large Systems

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IF.-\(: 10th 1"l'il'lllli;d ""odd \llIl1id .. FR(;. 1~ls7

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SIMPLIFIED DESIGN OF CONTROLLERS TO SOLVE THE ROBUST DECENTRALIZED SERVOMECHANISM PROBLEM FOR LARGE SYSTEMS A. F. Vaz and E. /)/"/"111/11/'111

11/

J.

Davison

F/I"III;III/ FlIglllI'I'I";lIg, 1 '11;"1"11;/\ "/ [11/'1111111. ()II/III;II

.\/'is /:H.

(;1/11111/11

Abstract: In [9], the Robust Decentralized ServomechanisllI Problelll (]{DSI') wa..~ forlllulated and soh'cd. In [12], [1:~i. and [15] parameter optimization techniques w('re developed to design controllers with dpsirable properties: dosed loop aSYlllPtotic regulation, fast res]>')[lSl', low channel interaction, integrit.y and tolerance to plant variations. The lIlethods of i12i and [13: produce excellent d,'signs - howl'\'"r they are computationall)' demanding: this lIlakes Ih('1Il illlpractical to apply directly to large scale system problems. In this paper, the llIultipar:lInder singular perturbation characl<'rization of lIlultilJlodclling introduced in fli] is used to d''''011l1'05(' the cOIllplex BDSI' into sIIlaller managl'ablp probl<'m5, Th,' resulting simplified desigll yields a controllC'r which behaves similarly to a controller that the complex design would produce. Keywords: Non linear programming; control of large scale interconnected systems; robust; servOIIl<'challislll problelll; d('centralized.

In tt·ou uc tion

Development

III control elllSineerinlS, one of the basic prob"'llIs which occurs is the design of realist.ic cOlllrollers to solve the lIIultivariable robust servomechanism problem, A mathematical model of the plant to be con trolled is assumed to be known to the designer. He then has the task of designing a t'ontroller with the following desirable properties: (a) th" closed loop system is asymptotically stable with a fast speed of 1'<'8pOnSe and without excessive high frequency oscillation; (b) asymptotic tracking and regulation occur for all reference signals and disturbance signals of a given class with low illteraet.ioll occurring between control challnels; (c) the controller gains have a reasonable magnitude; (d) the failure of a sensor or an actuator should not cause explosive instability to occur in the closed loop system; (e) the above properties should hold in the prescnce of parameter perturbation in the plant model. The solution to the above problem entails two distinct steps: the development of a synthesis theory and a design methodology. The synthesis thl'ory concerns itself with the retjuired structure and existence conditions to solve the servomechanism problem. The design methodology concerns itself with the numerical optimization of free parameters within a structural framework esta1,lished by the synthesis theory. Large scale systems are characterized by high system order and many inputs and outputs. Due to their size, they typically require decentralized information processing and some form of model reduction. A large class of practical systellls can be modelled as a composition of interconnected systems with slow and fast dynamics, where the slow dynamics are strongly coupled and the fast dynamics are weakly coupled. Examples of such systems are: power systems, mass-spring systems and multimarket economics. In [17], a multiparameter singular perturbation is employed to capture the multimodel nature of such systems. This is used to develop a modular synthesis theory in [25]. In this paper, the design methodology of [121 and [131 is adopted to the synthesis theory in [25].

115

A linear time-invariant system with a strongly coupled slow subsystem and weakly l'Oupled fast subsyslems is described by N

Xo =Aooxo+I;AojL) }~l

N

+ I;

Boj

It J

+-

Eow

]=1

N

Ei Xi

= Aio Xo -I A" Xi

+ I;

A,} x}

E,}

)~l

Yi

Yz~

= Cio Xo Ci~.to

:=..-=

C/i 1:,

Yi~ -:::-: ei

JI' +- iJ"ll, + E,w + Cii Xi + Di Iti + Fiw

=

(1 )

+ Fi~W

F/iw

I

y, - y,re f

(i = 1,2, ... ,N)

The vectors arc dimensioned as follows

x,ER

n ,

(i~1,2, ... ,N); Y}ER

r yljER ", e) ERr,

r

"

yloER r" ,

, Il}ERm, (j=1,2, ... ,N);

(2)

and N n

= I; n" i=O

N

r

= I;

N r}, and m =

j=l

I;

mj .

j=l

Subsystem i (i =O,I, ... ,N) has state Xi and disturbance w. Control agent j (j =1,2, ... ,N) is associated with: control input Il}, the output to be regulated y}' the regulation error e), the specified reference signal y ~ef, and auxirlary outputs YJo" J Yj' Yj~ and and" Yjj' It is assumed that yt) are the only outputs which can be measured. The following containment condition is assumed to hold: there exists Ti~ and T,~ so that Cio = Ti~ Ci~ and Cii=Ti~Ci~ (i=I,2, ... ,N). The information flow is said to be canonical if there exists T," such that A,o

=

T;"Ci~' (i=I,2, ... ,N).

(3)

.-\. F. Vaz and

116

E.J.

D,l\ison

Although (B) is not the most general form , no greater complexity is req uired when the inrormation flow is canonical (25 1. The compcasator (Bu) is only required when the fast dynamics canaot be stauilized uy constant output feedback (23 1.

If factoring condition (3) fails for some iE{1,2, ... ,N}, the information flow is said to be non-canonical. The small singular perturbation parameters Ei > 0 (i=1,2, ... ,N) represent time constant ratios between subsystem i and subsystem o. The states z, (i =1, 2, ... ,N) are "fast" since their derivatives are order l / Ei' which are large. We consider the case where E, (i = 1,2, ... ,N) are of the same order of magnitude, that is, there ex ist positive constraint I and L such that I~E .!EJ
Design Specifications

(i,j = 1,2, ... ,N). The small regular perturbation parameters (ij' io/=j, represent weak coupling between fast subsystems; they can be positive, negative or zero.

la this section, we consider a set of constraints that a practical controll'·r should sat isfy (121, [131, [151. Excessively larg e conlrol signals can cause sat uration effects. Such non lin ear ueh aviou r ca n be prevented if contro ller gaias are limit ed. Such a constraint can be implemented by requiring that

The disturbance vector wand the reference vectors y['! (i = 1,2 , ... ,N) are assumed to resp ectively satisfy

Vi = Ajvj and

V2 = A 2 v 2

w = E,v j

, = E 2v 2 1/,"! = G"

(4)

where (E"A,) and (E 2,A 2) are observable. ror nontriviality, it is assumed that a(Adea:' 1- and a(A2)ea:'+ where aC) denotes the spectrum of C) and a:' + denotes the closed right half complex plane. We also assume that Ea

Ej = dim(w),

rank

EN

IIA",A2i,A3"Koi,Koi/(.,1 1 < Mi IJr",I'2"K"K,11 < Mi for some M, > 0 (i = 1,2, ... ,N) , where IIXp ... ,Xkll
G: = {z I zEa:' , Re(z) < o[,

G,

G :=

I Re(z) 12 ~[ I Im(z) I- I Im(>-.') 1- 021} , and rank G = dimh)

(5)

where

GN where dimC) denotes dimension of vector (.). If this were not so, not all the disturbance or reference signal dynamics would affect system (1). It is desired now to solve the robust decentralized servomechanism probl em (RDSP) (91 for (1). Let the minimal polynomial of matrix A be denoted by mp(A). Let the lowest common multiple of a pair of polynomials (>, (s ) and (>2( s ) be denoted by LCM I(> j(s )'(>2(S )1. Associated with a pair of matrices A, and A2 is their spectral richness denoted by R(ApA2) and defined by R(A j,A 2):= {>-'I >-. is a zero of LCM (mp(A j),mp(A 2)J}

(6)

In (71 and [101, it is shown that a servocompensator must be modelled upon the spectral richness of the disturbance and reference signal models . In particular, a servocompensator for control agent i is of the form Ei = C i Ei + Di

ei

(7)

where a(C,) = R(ApA2). See (101 for details.

ryi = AhYi~+A2iC+A3it), = rliYii+rZiPi (i = 1,2, ... ,N)

(Ba)

(Bb)

Control agent i implements a controller of the form ui = Koi Yi~+Koit}i + i
The failure of an actuator or senso r can potentially cause explosive ins tab ility to occur in the closed loop system. To avoid such a scenario, the closed loop eigenvalues under such failur e conditions sho uld be constrained to lie in some safe region 5 of the complex plane. TlIe safe region 5 is defined by

5: = {z I zEa:',Re(z) < os} where Os is some small positive number selected by the designer.

In addition to the above, control agent i may contain a decentralized stabilizing compensator [271. In th e context of singular perturbations, such a controller has the following form

(iP,

In practice, system models cannot be identified exactly. SOllle plant d Yjlamics change with aging and wea r. Thus it is des irabl e that the closed loop eigenvalues also remain in the good reg ion G when certain plant parameters change from there nominal values. rormally one considers the plant model (1) to be a function of a vector of plant parameters p , that io, ADD = Aoo (p), Aoj = A oj (p) , Aii = Ai'(P), etc. where Pu represents the nomillal para 1Ileters. Let P be a family or plant parameters defined by P = {Pa,P ,,···, PI}· Thu s P induces a family of plant models (1). This formulatioll includes the sensor aad gain perturbatiolls of [131 as a spec ia l case.

(9)

Consider the output Y" which is composed of r, sensor outputs. If the (>th sensor of Y, fails , we set the (>th row of C,o' Cii , Di and Fii to zero. This process is repeated for multiple sensor failures . For sensor failures in Yi~ or Yi~' a similar zeroing process is used . Actuator failures are accommodated by an analogous process. Consider input ui which is composed of mi actuator inputs. If the {3th actuator of ui fails, we set the {3th column of B o,' B,i' and Di to zero. For multiple failures this process is repeated.

II i

Simplified lksi~n ot Controllers It can be seen that sensor and actuator failures induce changes in th e plant model (1) . Let f be a failure condition for a subset of th e actuators and sensors. Let F={f 0.1 1' .. ·.1 u}, where f 0 denotes that no actuators or sensors have fail ed. Thus we can consider matrices C io ' Gi .-, D i , F i .-, Ci~' Fi~ ' C/i, F/i, B ol ' and Bil to be fun ctions of f EF .

;7" = Ail Yi~ -I- Az, '7is -I- A3, Eis u" = 0,c l Kio y,~ -I- 0,-1K,o '7, s -I- 0 i-

I

w" = [0,- 1K,

Kio EIS

(12)

-I- ~,xs -I- W"

F,e: - 0,cl =i E, - Z, ]w

where

Closed Loop Dynamics

i= 1

When controller (8) is used on system (1) the followin g closed loop dynamics result

E IS : = BOI - AoiAi~IAio' C is : = Cio - Cn Ai;! AiD'

Ci~: = Ci~' D,s := D, - C"A,;IH", N

Es: = Eo - ~ A oiAi; IE" 1= 1

FIS : = F, - C"A, ; IE, Fl~ :== Fi~'

Z, : = (I -I- /(, C,aAi;1H,i

(10)

=,: = (/1"i - K,l'ii l\,)C,i A,;I,

0,: = J

-I-

=,B ii ,

~i : = _0,cl

=, A,o'

1\1, : = I'z, - f whe re

t l K,I\I,cll'li F,i ,

I, C,i(A" -I- B" /(, C,i tl Hi, K,.

Slow model (ll) with control (12) has the foll ow in g closed loop equations ~I

is = A sc Xs -I- [Bsc I

'71

es = C sc Is -I- [Dsc I

PI

Us = Ksc Is -I- [Ksc

[y~! ) Dscz] [y~! ) 01[y~! ] 8,,21

(13)

where

xs Let z be an output composed of the most integ rated output of each servocompensato r i (i = 1,2, ... ,N). Accordingly,

Eis

1:: s :=

ENs '7ls

uNs

'I Ns

wh e re

o D c :=

°°

DI

. .. ,

t° °

Using [25 ], the fast model corresponding to (10) is given

Di := di<1l, [(l,O, ... ,O), ... ,(l,O, ... ,O)]

by

repea ted r, times

Xi f

Using [25 ], the slow model corresponding to (10) is given by

£i

N

I , =A,x, -I-

~ Bi, uis -I- EsW , 1= 1

and

Yi, = Ci,Xi, -I- D is Uis -I- Fi,w,

Yi~ = Ci~Xs eis = Yis where

' ", ~ l" " ]'

+ Fi~W,

yr'!

(11)

for (i = 1,2, .. . ,N) and

r

i,! = Ai! c Iif where Ii!:= ~'f. )

(14)

A. F. \'az and E.

118

J.

Da\'json J defined by (15)

Performance Index for Controlled Plant In [121 and [131, a performance index is formulated that reflects the speed of response and the in t eraction of outputs in the RDSP. This performance index is used along with hill climbing techniques [81, [14 1, [151 to find an optimal set of controller para meters. This is achieved by first parametrizing the closed loop res ponse in terms of unknown controller parameters. The performance index is then evaluated for different parameter values using a hill climbing technique t o find an optimal set. The parameter search is restricted to the region that meets the design constraints on controller gains, closed loop spectrum, tolerance to model variations, and integrity. In this section we formulate the performance index. When system (1) is used with controller (9), th e closed loop dyn a mics are giv en by (lO) . By using the res ul ts of 1131, we have the following performance ind ex

K."K, ,, K.,

r" ,r" ,K" K. The constrained full order design problem has the additiona l property that it is subject to cons traints , for example:

Gain Cons traint I[Ali ,A2i,AJ,,I(o,,Koi,Koi ll

< Mi

Spectral Constrain t

ACll AC12] a[ CC Ac21 Ac22 Tolera nce Cons traint

11

12]

Ac Ac a[ Ac21 Ac22

p EP

C C

Integrity Constraint ACll ACI2] a[ Ac21 Ac22 f

where

EF

CS

This non linear programming problem alw ays has a solution prov id ed an initial starting point for the problem exists. For the case when the pla nt is open loop stable and the disturbances and reference signals are of the polynomial type, the centralized tuning method [11 1 can be used sequentia lly to giv e an initi al starting point. We term the following parameter optimization problem to be the simplified design problem

J defined by (16)

and rAdl ACI2]T P

lAc21 _

+prAC11

Ac22

lAc21

K."K",K.,

AC12] = Ac22

r" ,r" ,K" K; The constrained simplified de sign problem has the additional property that it is subject to constraints, for example:

[D?QD~+KCIRKcl Kc~RKcll Kc2 RKcl

Kc~RKc2

and Q > 0 and R > 0. The parameters (ag,at,ai) are used to weigh t th e emphasis of the performance ind ex upon initial conditions, disturbance rejection and tracking respectively .

Gain Constraint

In an an a logous manner we can derive a performance ind ex for the slow mod el. Wh ere system (10) is used with control (11) we use th e following simplified performance index

Spectral Constraint

J, = agtrP, + attr(T~P, T l , )

+ a 22 tr(T[,P, T 2, )

IIAli ,A2i,AJ"Koi ,Koi,Koi ll Ilrli,r2i,Ki,Kill

< Mi

< Mi ;

a(A ,, ) CC, a(A ifc / €d CC; Tolerance Constraint

a(A,,)lp EPCC , a(Aifc / €i)lp EPCC; (16)

Integrity Constraint

a(A,,)lf EFCS, a(A,/c / €i)lf EFCS,

where

A,, [T l, T2 ,1-[ Tl ' T 2,I

[~l ~J

= -[ E" lEI E,,2GE2 1 and

for (i = I,2, ... ,N). The above constraints have the advantage that they can be verified in a modular fashion. Before stating our first main result, we make the following definition. Define 11i := (Ali ,A2i ,AJi,Koi ,Koi,Koi ,r li,r 2i ,Ki ,Ki ), for (i = I,2, ... ,N).

Pal'ameter Optimization Problem A solution to the flDSP with a fast speed of response and low interactioll is obtained by solving for controller parameters in the following optimization problem, which we term the full order design problem

Theorem 1 (Simplified Parameter Optimization) Suppose 11i (i = I,2, .. .,N) satisfy the gain constraint, spectral constraint, tolerance constraint, and integrity constraint for the simplified design problem. There exists an / > 0 so that for 'all O<€j < €', I €ij I
Simplified Design of COlltrollers (a)

'If, (i=1,2, ... ,N) satisfy the same constraints in the constrained full order design problem.

(b)

J('If) , ... , 'lfN) = J,(1I'), . .. , 1I'N)

(c)

IfJ,(1I')', ... ,1I'N)= JOP

rr,

+ OV)

J,(1I'), ... ,1I'N)

('=).2 ,... ,N)

I

then

min

=

J(1':)', ... ,1I'N) + 0V)

o Theorem 1 gives the control system designer a mechanto reduce the size of matrices that he must handle. Th)s reduces the number of computations and the numerical difficulties. To further alleviate the computational burden, we must decrease the number of parameters to be optimized. This can be done for the special case when a modularity condition holds. The modularity condition holds if the following are satisfied.

isu:

1. 2.

Ip EP =

T,OCj~

Conclusion The problem of designing a decentralized controller for a large scale system has been considered. Our design method produces controllers with the following desirable properties: closed loop stability, asymptotic regulation, fast response, low interaction, integrity and tolerance to plant variations. Rather than deal with the dynamics of the plant as a whole, we extract a lower order slow model and a set of decoupled fast models. This allows us to avoid repeatly solving high order Sylvester equations and eigenvalue problems. Furthermore, if our modularity condition holus, we need only optimize a subset of the COil troller parameters. Th us we can design contro llers for significantly more comp lex systems, than would normally be realistic to consider. Some numerical e;\amples can be found in [26 ). References

Ej = O, Fj~ = O, and F,~ = O (i = 1,2, ... ,N) There exists T,o (i = I,2, ... ,N) such that Aja

[1)

R.H. Bartel and G.H. Stewart, "Algorithm 432, solution of matrix equation AX + XB = C" , Comm. Assoc. Comp. Mach., Vol. 15, pp. 820-826, 1972.

[2)

P. Chemouil and A.M. Wahdan, "Output feedback cont rol of systems with slow and fast modes", J. Large Scale Systems, Vol. 1, pp. 257-264, 1980.

[3)

J.H. Chow and P .V. Kokotovic, "A decomposition of near optimum regulators for systems with slow and fast modes", IEEE Trans. Auto. Control, Vol. AC-21, pp. 701-705, Oct. 1976.

[4)

J.H. Chow, et al., Time Scale Modeling of Dynamic Networks, Lecture Notes in Control and Information Sciences, 46, Springer Verlag, New York, 1982.

[5)

S.D. Conte and C. deBoor, Elementary Numerical A nalysis: A n A 19orithmic Approach, McGraw-Hill Book Co., New York, 1980.

[6)

E.J. Davison and F.T. Man, "The numerical solution of A f Q +QA =- C", IEEE Trans. Auto. Con· trol, Vol. AC-23, pp. 448-449, 1968.

[7)

E.J . Davison, "The output contro l of linear time invariant multi variable systems with unmeasurable arbitrary disturbances", IEEE Trans. Auto. Contro l, Vol. AC-17, No. 5, pp. 621-630, Oct. 1972.

[8)

E.J. Davison and P. Wong , "A robust conjugate _ gradient algorithm which minimizes L -functions", Automatica, Vol. 11, pp. 29i-308, 1975.

[9)

E.J. Davison, "The robust decentralized cont rol of a general servomechanism problem", IEEE Trans. Auto. Control, Vol. AC-21, No. 1, pp. 14-24, Feb. 1976.

I pEP ! EF

When the above modularity condition holds, we make the following definitions

K j.: = 6,c)K ja - 6 j-)2 j T jO

K"

:=

6 j- ) K,o

K j, : = 6," ) K,a which gives Uj,

= K j,

Yj~

+ K j, 1]js + K j, ~j,

.

Before stating our second main result, we make the following definitions. Define 1fis :== 1I'if: =

(All ,A2i ,A3i ,Kis ,Kt's ,Kis ) , (f)j,f2j,Kj,Kj),

*i~: = (Al:· ,A2:" A3·i,Ko'i,~:· ,ko:· ,rli , r 2i ,Kz ,K,), where

Ka: := 6, K,;

+ 2, T,o

~',: = 6,K,~,

Ka',: = 6jkj~

for (i =1,2, ... ,N). Theorem 2 (Modular Parameter Optimization) Suppose the modularity condition holds and 1':" (i = 1,2, ... ,N) cause A" to satisfy the spectral constraint, tolerance constraint, and the integrity constraint of the simplified design problem. For all 1':if (i = 1,2, ... ,N) which cause A j ! c / E, to satisfy the same constraints, there exists an E' > O so that for all O< E, < E', IEj} I < E' (i,j=I,2, ... ,Nj irfj) , the following results hold. (a)

(i=1,2, ... ,N) satisfies the same constraints as in the constrained full order design problem.

1':j

(b) J(1':), .. . ,1':N)=J,(1I')" ... ,1':N,) + O(l') (c)

If J, (1':)'" ... ,1I'N,) =

min

J,(1':)" ... ,1I'N,)

(' = ),2, ... ,N)

then

JOpl

= J(ir)', ... ,ir~)

ll9

+ 0(£'),

o

[10) E.J. Davison, "The robust control of a servomechanism problem for linear time invariant multivariable systems", IEEE Trans. Auto . Control, Vol. AC-21, No. 1, pp. 25-34, Feb. 1976. [11 ) E.J. Davison and W. Gesing, "Sequential stability and optimization of large scale decentralized systerns", Automatica, Vol. 15, pp. 307-324, 1979. [12) E.J. Davison and I.J. Ferguson, "The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods", IEEE Trans. Auto. Control, Vol. AC-26, No. 1, pp. 93-110, Feb. 1981. [13) E.J. Davison and T.N. Chang, "Decentralized Controller Design Using Parameter Optimization Methods", Control: Theory and Advanced Technol· ogy, Special Issue on Large Scale and Complex Systems, Vol. 2, No. 2, pp. 131-154, June 1986.

I~()

.-\. F. \'a l alld L.J. D,l\isoll

114) W. Gesing and E .J. Davison, "Improvements on a robust conjugate-gradient algorithm which minimizes L-functions" , A utomatica, VoL 14, pp . 515-516, 1978. 115) W. Gesing and E.J. Davison, " An exact penalty function algorithm for solving general constrained parameter optimization problems", A utomatica , VoL 15, pp. 175-188, 1979. 116) A.H. Haddad and P.V. Kokotoyic , ":-.Iote 011 sillgular perturbation of lin ea r state regulators", IEEE Tran s. Auto. Con trol, Vol. AC- 16 , No. 3, pp. 279-281 , June 1971. 117 ) H.K. Khalil and P.V. Kokotovic, "Control strategies for decision makers using different models of the same system", IEEE Traits. Auto. Control, Vol. AC-23, No. 2, pp. 289-298, April 1978. 118) H.K. Khalil and P.V. Kokotovic, "Decentrali zed stabilization of systems with slow and fast modes", J. Large Scale Systems, Vol. 1, pp. 141-148, 1980. [19 ) P.V. Kokotovic , RE. O' Mall ey and P. Sannuli, "Singu lar perturbations and order reduction in control theory - an overview", Automaiz'ca, Vol. 12, pp. 123-132, 1976. 120 ) P.V. Kokotovic, "Subsystems, t imes scales, and multimodeling", Au!omatica, VoL 17, No . 6, pp. 789795 , 1981. 121 ) P.V. Kokot.ovic, "Ap plications of singular perturbation techniques to c.ontrol problems", SIAM Review, Vol. 26, No. 4, pp. 50 1-550, Oct. 1984. 122 ) B.N. P a rlett , The Symmetric Eigenvalue Problem, Prentice Hall , In c., E nglewood CliITs, N.J. , 198D. 123 ) B. Porter, " Singular perturbation methods in t.he design of full ord er observers for multivariable lin ear systems" , Int . 1. Control, Vol. 26, No. 4, pp. 589594,1977. 124) V.R. Saksena, J. O'Reilly and P.V. Kokotovi c, " Singul a r perturbat ions and time-sca le methods in contro l theory: survey 1976-1983" , Automatica, Vol. 20, No. 3, pp. 273-293, 1984. 125 ) A.F. Vaz and E.J. Davison, "Time scale based modular synthesis of controllers to solve t he robust decentralized servomechanism problem" , Systems Con trol Report No. 8607, Dep t . of Electrical Engineering, University of Toronto, May 1986, 10th IFAC World Congress, Munich , July 1987.

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